## Introduction

The turnover frequency (TOF) of a catalytic reaction is the traditional measure of the efficiency of a catalyst, expressed as the number of cycles performed per time unit and catalyst concentration. The TOF is routinely available in experimental chemistry, but until recently, there was no simple and straightforward method to calculate it from the conventional energy profiles obtained by means of computational chemistry techniques. Such a link is important as the interplay of theory and experiment can be extremely fruitful in catalysis.

The energetic span model developed by Kozuch and Shaik1–6 has the potential to interweave computational and experimental chemistry. It allows to calculate the TOF of a catalytic reaction from the energy profile4 by the following eq. (1), which gives the energy-representation (E-representation) of the TOF:

Here Δ*G*_{r} is the energy of the global reaction. *T*_{i} and *I*_{j} are the calculated free energies of each transition state and intermediate, respectively. δ*G*′_{ij} is either Δ*G*_{r} or zero, according to the position of *T*_{i}*vs. I*_{j} in the specific term of the summation, as specified in eq. (2):

In eq. (1), Δ (the numerator of the TOF formulation) corresponds to the driving force of the reaction; whereas *M* is the kinetic resistance. In analogy to Ohm's law, a catalytical flux law can be established: TOF (the catalytic current) is equal to Δ (the catalytic potential) divided by *M* (the kinetic resistance to catalysis).3 The Δ and *M* terms were introduced by Christiansen7 to define the TOF as a function of kinetic rate constants (the k-representation)4 in a steady state regime. The Δ and *M* terms in the E-representation can be derived from Christiansen's formulation by applying Eyring's transition state theory.2

Using the E-representation in eqs. (1) and (2), the computational chemist can calculate TOFs and compare the efficiency of different catalytic systems from computational data, and therefore suggest the most suitable mechanism, catalyst, substrate, or solvent for experimental examination. *Vice versa*, experimentalists can test the computational study against their experimental TOF values, and thereby provide feedback to the computational chemist on the accuracy of his or her calculations.

Herein, we apply the energetic span model to a recent computational study regarding the hydroamination of ethylene with ammonia to form ethylamine, catalyzed by a rhodium pincer complex.8 In addition, we present the AUTOF program that allows the user to apply the complete model in a black box fashion.